Event-triggered fixed-time adaptive fuzzy control for state-constrained stochastic nonlinear systems without feasibility conditions

The problem of event-triggered fixed-time control for state-constrained stochastic nonlinear systems is discussed in this article. Different from the barrier Lyapunov function (BLF)-based and Integral BLF-based schemes that rely on feasibility conditions (FCs), by introducing the nonlinear state-dependent functions, the asymmetric time-varying state constraints are handled without FCs. Combined with the fixed-time stability theory and the dynamic surface control technique with fixed-time filter, the fixed-time stability in probability of the closed-loop system is ensured and the problems of “explosion of complexity” and “singularity” are overcome. Furthermore, the novel fixed-time error compensation signals are designed to compensate the filtering errors, and the event-triggered control technique is used to save network resources. Simulations also illustrate the effectiveness of the proposed method.


Introduction
As the requirements of control performance, physical limitations or security considerations, the research of adaptive control for state-constrained systems has attracted a lot of attention, and many significative results have been presented. For instance, model predictive control [1], reference governors [2,3], and set invariance notions [4][5][6]. The authors of [7] first proposed an integrator backstepping scheme based on BLF to deal with multi-state constraints. Then, many BLFor IBLF-based schemes for state-constrained systems with different forms were presented, see [8][9][10][11][12][13][14] and references therein. However, one drawback of the BLF-or IBLF-based schemes is that the virtual control function needs to satisfy the FCs, i.e., the virtual control function needs to be limited a pre-given constrained region. The existing solution is to find a set of design parameters to satisfy the FCs. Obviously, this is a very complicated process. Moreover, as stated in [15,16], the parameters that satisfy the FCs may not exist when the states are limited to a small set. To this end, the new schemes based on NSDF were presented [17,18], under which the FCs can be completely removed.
As is known to all that the stability time of the system is also an important index to evaluate the controller.
Nevertheless, most of the above schemes can only ensure asymptotic stability. To obtain a finite convergence time, the finite-time stability theory was first proposed [19]. On the basis of this theory, many finite-time control schemes were proposed for state-constrained systems [20][21][22][23]. One disadvantage of these methods is that the setting time is always dependent on the initial value of the system. It means that the convergence time will be longer as the initial state deviates from the equilibrium position. Then, the authors of [24] first presented the fixed-time stability theory, under which the convergence time can be predicted in prior and independent of the initial state. Then, many adaptive fixedtime control (FTC) schemes were proposed [25][26][27]. The authors of [28] extend the fixed-time stability theory to stochastic systems.
As the order of the system goes up, the problems of "explosion of complexity" and "singularity" caused by repeated differentiations of virtual controllers frequently appear in backstepping design procedure. To solve these problems, the DSC technique with a firstorder linear filter was first presented in [29]. Then, many DSC schemes with different filters were proposed. To name a list, the DSC schemes with linear filter [30], nonlinear filter [31,32], finite-time command filter [33,34] and fixed-time filter [35] were proposed for nonlinear systems with different forms.
In addition, most of the control schemes for stateconstrained systems were constructed under the timetriggered control (TTC) framework, under which the output of the controller is applied to the system continuously, whether it's needed or not. Clearly, this will produce a lot of redundant data and increase the communication burden. To this end, the ETC technology was proposed in [36][37][38]. Nevertheless, a limitation of the above works is that the assumption of input-to-state stability is needed. To relax this restriction, an effective ETC method with relative threshold technology was presented in [39]. Then, the method was applied to the time-delay systems [40] and stochastic systems [41,42]. On basis of the finite-time stability theory and the fixed-time stability theory, two event-triggered finite-time controllers [43,44] and an event-triggered fixed-time controller [45] were designed.
Inspired by the previous work, it can be seen that most of the control schemes for state-constrained stochastic systems were constructed by the conventional backsteeping technology under the TTC frame-work. There are two drawbacks in the above methods: (1) As the order of the system goes up, the problems of "explosion of complexity" and "singularity" caused by repeated differentiations of virtual controllers frequently appear in backstepping design procedure; (2) The TTC schemes will produce a lot of redundant data and cause the waste of network resources. Therefore, a significant problem arose naturally: for asymmetric time-varying state-constrained stochastic systems, can we design an event-triggered fixed-time controller without the FCs? As far as we know, this problem has not been solved until now. The advantages of this note are given as follows (i) Compared with the BLF-or Integral BLF-based schemes that rely on the FCs in [20][21][22][23], under which the virtual controllers need to be limited a pre-given constrained region. In this note, by introducing the NSDFs that purely rely on the constrained states, the state constraints are handled directly and the FCs are removed. Moreover, note that the state constraint functions considered in this paper are time-varying asymmetric, but the ones in [20][21][22][23] are limited to symmetric constants. (ii) Unlike the finite-time command filtered control methods [23,33,34], by combining with the fixedtime stability theory and the DSC technology with fixed-time filter, the fixed-time stability in probability of the closed-loop system can be ensured. Meanwhile, the problems of "explosion of complexity" and "singularity" are avoided. In addition, ETC strategy is used to save network resources. Compared with the scheme in [45], the novel FTECSs are designed to compensate the filtering errors. Moreover, the considered systems contain stochastic disturbances and asymmetric time-varying state constraints, which are more general and can enlarge the practical application range.
The remainder of this note is addressed as follows. Section 2 gives the relevant preliminaries. The design procedure and analysis are shown in Sect. 3. The Simulations are shown in Sect. 4. In Sect. 5, a conclude is given.

Key definition and lemmas
Definition 1 [28] Given a stochastic system as follows V (ζ ) is a positive definite function, the differential operator L of V (ζ ) is defined as Lemma 1 [28] Let V (ζ ) : then, the solution of system (1) is fixed-time stable in probability (FTSP), and the setting time T satisfies then, the solution of system (1) is FTSP, and the setting Proof By introducing a constant η ∈ (0, 1), then inequality (4) can be rewritten as or Based on Lemma 1, one can obtain that the solution of system (1) is FTSP, and the setting time satisfies Based on Lemma 1, one can obtain that the solution of system (1) is FTSP, and the setting time satisfies Combing case 1 with case 2, we can obtain the conclusion of Lemma 2.
Remark 1 For convenience, in this note, let γ 1 = 3 4 , γ 2 = 2. Lemma 3 [13] H (χ ) denotes an unknown nonlinear function, for a given accuracy > 0, there exists a fuzzy logic system (FLS) W T S(χ ) such that where χ, W denote input vector and weight vector, respectively. (χ) denotes a bounded approximation error, i.e., is a basis function vector with l > 1 being the number of the fuzzy rules, and s i (χ ) denotes a Gaussian function of the following form where μ i and η i denote the center vector and spreads of the Gaussian function, respectively.

Problem statement
The considered system is expressed as whereζ i = [ζ 1 , ζ 2 , . . . , ζ i ], u(t), y denote system state, system input and system output, respectively. f i (·) and h i (·) ∈ R r denote unknown nonlinear function and function vector, respectively. g i (·) denotes a known continuous function. ω ∈ R r is an independent standard Brownian motion. All the state vari- The aim of this article is to construct an effective controller u(t) to ensure that the whole variables of system (16) are fixed-time bounded in probability (FTBIP), and that the output y tracks the desire trajectory y r within a fixed-time interval. In the meantime, the time-varying asymmetric state constraints are not violated.
Assumption 1 [41] The desired signal y r and its derivativeẏ r are continuous, and y r satisfies : −F 11 (t) ≤ y r ≤ F 12 (t).

Controller design
To cope with the state constraints, the NSDF [17] is introduced as follows From (17), one has where Then system (16) can be rewritten as where Remark 2 Compared with the BLF-or Integral BLFbased schemes that rely on the FCs in [20][21][22][23], under which the virtual controller needs to satisfy n). By introducing the NSDF, the FCs are removed. Moreover, the state constraint functions in [20][21][22][23] are the simpler case of symmetric constants, while the case of asymmetrical time-varying state constraints are considered in this paper.
Remark 3 Note that the NSDF is also used in [17], but the method in [17] only ensures that the system is stable as time tends to infinity. In this note, by combining with the fixed-time stability theory and the DSC technology with fixed-time filter, the fixed-time stability in probability of the closed-loop system can be ensured. At the same time, the novel FTECSs are designed to compensate the filtering errors. In addition, the systems considered in [17] are limited to deterministic strict-feedback nonlinear systems. Let . Define the following coordinate transformation: where s i,c is the output of the fixed-time filter with α i−1 as the input, α i is the virtual control function. The fixed-time filter [35] is designed as follows where ξ i,1 = s i+1,c ,ξ i,1 =ṡ i+1,c , φ 11 , φ 12 , φ 21 , φ 22 denote design parameters, 1 and 2 denote positive design constants, ν i > 1, ν i = iν − (i − 1) and ν = ν 1 . From [35], with the above filter, |s i+1,c − α i | ≤ κ i in a fixed time.
To reduce the influence of filtering errors, the novel FTECSs are designed as follows where l i1 , l i2 denote positive design parameters.
Remark 5 Compared with the existing DSC methods with finite-time command filter in [33,34], by introducing the fixed-time filter, the proposed method can not only eliminate the problems of "explosion of complexity" and "singularity", but also ensure the fixedtime stability in probability of the closed-loop system. Moreover, the influence of filtering errors is reduced effectively by introducing the novel FTECSs.
Let the compensated tracking error be e i = z i − i .
. . , n. α i and˙ i are constructed as: where ε i , k i1 , k i2 , r i , σ i , c i are positive design parameters.
The event-triggered controller is defined as follows: where t k denotes the event-triggered time with k ∈ Z + , ρ, q ∈ R + , λ ∈ (0, 1) andh > q/(1 − λ) . (29)- (31), we can see that u(t) is a constant when t ∈ [t k , t k+1 ). Compared with the TTC schemes, the ETC technology can save communication resources effectively. In addition, the relative threshold strategy is introduced, which is more flexible than the fixed threshold strategy.

Main results
The main results are concluded by the following Theorem.
Theorem For system (16) with Assumptions 1-2, let the actual control input be designed as (30), the virtual control functions be designed as (25), (26), and the update law be designed as (28). The proposed scheme guarantees that the whole variables of system (16) are FTBIP, and that the output y tracks the desire trajectory y r within a fixed-time interval. In the meantime, the time-varying asymmetric full-state constraints are not violated without involving the FCs.
Based on (74)-(76), we have where K 2 =K 2 2n . According to Lemma 2 and the definition of LFCs, we can obtain that e i and˜ i are FTBIP, and the setting time satisfies: (25)- (27), one can obtain that α i and i are also FIBIP. Since In what follows, we will show that i is FTBIP.
Step n+1 Select the following LFC Then, one has According to [35], there exists a given positive constant κ i such that s i+1,c − α i ≤ κ i is hold in a fixed time T i2 . Then, one has where According to Lemma 1 and the definition of V n+1 , it is obtained that i is FTBIP, and the setting time satisfies:T 3 = 2 L 2 + 1 L 1 . Since e i = z i − i , it is easy to obtain that z i is FTBIP, and the setting time satisfies: T = max{max{T i2 } + T 3 , T 1 }. Then s i is FTBIP, from (17), we can obtain that ζ i is FTBIP. According to the above analysis, the conclusion that the solution of system (16) is FTSP can be obtained. From (17), one has −F i1 (t) ≤ ζ i ≤ F i2 (t), i.e., the time-varying asymmetric full-state constraints are not violated. Furthermore, as z 1 satisfies , thus, the tracking error z * 1 = ζ 1 − y r can be limited to a small residual set within a fixed-time interval by choosing the appropriate parameters.
From Figs.1 and 2, it can be seen that the proposed scheme can achieve a good tracking performance, and the asymmetric time-varying state constraints are not violated. Figures 3, 4 and 5 express the profile of the tracking error z * 1 , the estimated parametersθ 1 ,θ 2 and the control signal u(t), respectively. The time interval of triggering events is shown in Fig. 6.

Conclusion
In this article, the problem of event-triggered fixed-time control is studied for state-constrained stochastic systems. By introducing the NSDFs, the state constraints  novel FTECSs are designed to compensate the filtering errors, and the ETC technique is used to save network resources. Theoretical analysis and experimental simulations verify the effectiveness of the proposed scheme. To extend the scheme to multi-agents systems with stochastic disturbances is also the direction of our future efforts.